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Use the method of multiple scales, treating $t$ and $\tau =t/ϵ$ as independent variables, so differentiation becomes ${\partial }_t+{ϵ}^{-1}{\partial }_{\tau }$. Write $L={\partial }_x^2-{\ell }^2$ and average over a fast period $2\pi /|\omega |$, with $\omega \ne 0$. The fluctuations can depend slowly on $x,t$ as well as periodically on $\tau$. The ${ϵ}^{-1}$ balance in the poloidal magnetic field potential equation isIntegrating and imposing zero fast average determinesAt order one, the toroidal magnetic field equation is ${\partial }_t\overline{B}+{\partial }_{\tau }\tilde{B}=\Omega {\partial }_x(\overline{A}+\tilde{A})+L\overline{B}$. Its average and fluctuating parts give, respectively,Consequently the zero-average fluctuation isThe order-one averaged poloidal magnetic field potential equation retains the correlation between the fluctuating alpha effect and the fluctuating toroidal magnetic field. Although alpha has zero average, its product with the magnetic fluctuation does not:Thus the rapidly fluctuating alpha effect produces the leading averaged equationsA higher-order zero-average correction to $A$ accommodates the remaining fast order-one terms; it does not alter this leading averaged balance. These equations describe the small-$ϵ$ limit, rather than an exact finite-$ϵ$ identity.

For a Fourier mode, let $(\overline{A},\overline{B})=(a,b)e^{ikx+\sigma t}$ and put $d=k^2+{\ell }^2$. Then $(\sigma +d)a=-ikCb$ and $(\sigma +d)b=ik\Omega a$. Their determinant gives the dispersion relationHere $\Omega C={\Omega }^2{\alpha }_0^2/(2{\omega }^2)\ge 0$, so both growth rates are real. For a prescribed nonzero wavenumber, growth requires $\Omega C>(k^2+{\ell }^2{)}^2/k^2$. If real wavenumbers can be chosen freely, the largest growth rate occurs at $|k|=\sqrt{\Omega C}/2$ and is $\Omega C/4-{\ell }^2$. Hence growing averaged modes exist precisely whenA bounded or periodic spatial domain instead requires an admissible wavenumber satisfying the preceding mode-by-mode inequality.

The physical feedback differs from an ordinary Parker dynamo wave. A constant alpha effect couples $B$ directly into $A$, while the Omega effect differentiates $A$ once, producing the factor $ik\alpha \Omega$ responsible for an oscillatory phase. Here the fast alpha effect first creates a potential fluctuation in temporal quadrature; shear creates a magnetic fluctuation correlated with alpha. The averaged feedback is consequently proportional to $-{\partial }_x\overline{B}$. Combining its spatial derivative with the derivative in the Omega effect gives the real positive factor $\Omega C{k}^2$. In fact $({\partial }_t-L{)}^2\overline{B}=-\Omega C{\partial }_x^2\overline{B}$. The resulting growing fields have fixed spatial phase, so the leading averaged pattern grows without travelling. Fast within-period oscillations of the original fields remain present.